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**Additional resources for Computational quantum mechanics for materials engineers: the EMTO method and applications**

**Example text**

4) becomes [47, 80] wR sR R v0 = R 2 rR v(r)dˆrR drR 3 − s3 )/3] [4π(wR R . 56) When sR → wR , the above expression reduces to v0 → v(r)dˆrR rR =sR 2 4πs R R R s2R R = vR (sR )s2R . 1) are given in terms of the spherical symmetric part of the full-potential, which can be computed eﬃciently and with high accuracy. 4). e. 5). Since both of these potentials have long range, they should be grouped in such a way that at large distance the negative and positive terms cancel each other. Usually, this is done by dividing ve (r) + vH (r) into components due to the charges from inside and from outside of the cell at R.

9)), a multi-center form for the charge density can be obtained. Although, this multi-center expression gives a highly accurate charge density in the entire space, its application in the Poisson equation or the total energy functional is very cumbersome. 9). 33). This expression is valid inside the potential spheres. Nevertheless, due to the kink-cancellation equation, the one-center expression remains valid for rR > sR as well, if the a normalized partial wave NRl ( j ) φRl ( j , rR ) is replaced by the backward extrapolated free-electron solution ϕaRl ( j , rR ).

3). The ﬁnal expression can be cast into the following form: 7 In practice, in order to be able to compute the density gradients and eventually the higher order density derivatives, the partial densities should in fact be deﬁned inside a sphere which is slightly larger than the sphere circumscribed to the Wigner−Seitz cell. 49) where CLL L are the real Gaunt numbers (Appendix B). For the radial functions the following notation has been introduced: ⎧ a N (z)φRl (z, rR ) if l ≤ lmax and rR ≤ sR ⎪ ⎨ Rl a ϕaRl (z, rR ) if l ≤ lmax and rR > sR .